Here's another game-related problem. Consider the game of "othello" (aka "reversi" which has the same rules but the other start position; 8x8 board, bicolored disks, start with 2 disks of each color in the central 4) Q1. what is the maximum "mobility" i.e. number of legal moves, in an othello position? Q2. same question, but position must be reachable from gamestart position via legal move sequence. (Hence the answers for Q2 might differ for reversi vs othello.) Q3. What is the shortest possible "wipeout" game (ends when one side has no disks)? (Again, answers for Q3 might differ for othello vs reversi.) Q4. What is the shortest possible nonwipeout game (ends when both sides have no moves)? (Again, answers for Q4 might differ for othello vs reversi.) Q5. Sid Cox conjectured any fully-filled-board (final) position reachable from the reversi start position, also is reachable from the othello start position. Q6. Aubrey de Grey showed that the maximum possible number of times a disk could be flipped is given by the following table, in an othello game: 0 1 2 3 3 2 1 0 1 17 18 17 17 18 17 1 2 18 20 20 20 20 18 2 3 17 20 23 23 20 17 3 3 17 20 23 23 20 17 3 2 18 20 20 20 20 18 2 1 17 18 17 17 18 17 1 0 1 2 3 3 2 1 0 and all these upper bounds are tight EXCEPT perhaps the four 17s located adjacent to the corners, for which he proved a lower bound of 16. Open question: 16 or 17? A1. As opening bids, 24 is achievable using rows BWooWBWo. Better: 33 is achievable: oooooooo WWWWWWWo BWoBBBWo oooooBWo oWWWoBWo oWBWooWo oWWWoWWo oooooBWo I suspect 40 is a (probably weak) upper bound. Neither of these positions is reachable. A3. In Martin Gardner's Scientific american column (April 1960?), for Reversi, 1. d5 2. e5 3. d4 4. e4 (these 4 are the opening position, now for genuine moves:) 5. f4 6. e3 7. f2 8. c4 9. b4 10. c5 11. d6 is an 11-0 wipeout. This final position is not reachable from the othello start. For othello, 1. d5 2. e5 3. e4 4. d4 (these 4 are the opening position, now for genuine moves:) 5. e6 6. f4 7. e3 8. f6 9. g5 10. d6 11. e7 12. f5 13. c5 and 5. d3 6. c5 7. e6 8. d2 9. c4 10. f5 11. c6 12. b5 13. d6 14. d7 (13-0 and 14-0 wipeouts) were found by Manabu Maruo and David Haigh. I think these are claimed to be proven shortest possible. A4. Aubrey de Grey found an othello game ending with 15-4 score which he conjectures is shortest. http://www.britishothello.org.uk/killball.pdf -- Warren D. Smith http://RangeVoting.org